Introduction

Directional statistics is a branch of statistics dealing with observations that are directions. In most cases, these observations lie on the circumference of the unit circle of R $ {\mathbb{R}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119472/5ff6ea01-2f47-4320-8bbe-e544021cff79/content/inline-math1_1.tif"/> ² (one then speaks of circular statistics) or on the surface of the unit hypersphere of R $ {\mathbb{R}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119472/5ff6ea01-2f47-4320-8bbe-e544021cff79/content/inline-math1_2.tif"/> ^p for p ≥ 3 (implying the terminology of spherical statistics) ¹ . Data of this type typically arise in meteorology (wind directions), astronomy (directions of cosmic rays or stars), earth sciences (location of an earthquake’s epicentre on the surface of the earth) and biology (circadian rhythms, studies of animal navigation), to cite but these. The key difficulty when dealing with such data is the curvature of the sample space since the unit hypersphere or circle is a non-linear manifold. This can readily be seen on a very simple example. Imagine two points on the sphere of R $ {\mathbb{R}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119472/5ff6ea01-2f47-4320-8bbe-e544021cff79/content/inline-math1_3.tif"/> ³ and consider their average. This point will in general not lie on the sphere. This reasoning of course extends to several points on the sphere, entailing that the basic concept of sample mean needs to be adapted in order to yield a true mean direction on the sphere. Thus, the wheel has to be reinvented for virtually every classical concept from multivariate statistics.